Abstract

We investigate the Gouy phase shift for full-aperture waves converging to a focal point from all directions in two and three dimensions. We find a simple interpretation for the Gouy phase in this situation and show that it has a dramatic effect on reshaping sharply localized pulses.

© 2012 Optical Society of America

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References

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  1. L. G. Gouy, C. R. Hebd. Séances Acad. Sci. 110, 1251 (1890).
  2. L. G. Gouy, Annal. Chim. Phys. 24, 145 (1891).
  3. T. D. Visser and E. Wolf, Opt. Commun. 283, 3371 (2010).
    [CrossRef]
  4. S. Feng and H. G. Winful, Opt. Lett. 26, 485 (2001).
    [CrossRef]
  5. U. Leonhardt, New J. Phys. 11, 093040 (2009).
    [CrossRef]
  6. Y. G. Ma, S. Sahebdivan, C. K. Ong, T. Tyc, and U. Leonhardt, New J. Phys. 13, 033016 (2011).
    [CrossRef]
  7. S. W. Hell, S. Lindek, and E. H. K. Stelzer, J. Mod. Opt. 41, 675 (1994).
    [CrossRef]
  8. T. Tyc and X. Zhang, Nature 480, 42 (2011).
    [CrossRef]
  9. U. Leonhardt and T. G. Philbin, Phys. Rev. A 82, 057802 (2010).
    [CrossRef]
  10. M. Born and E. Wolf, Principles of Optics (Cambridge University, 2006).
  11. G. B. Arfken, H. J. Weber, and F. E. Harris, Mathematical Methods for Physicists (Harcourt, 2005).
  12. U. Leonhardt and S. Sahebdivan, J. Opt. 13, 024016 (2011).
    [CrossRef]
  13. P. Saari, Laser Phys. 12, 812 (2002).

2011 (3)

Y. G. Ma, S. Sahebdivan, C. K. Ong, T. Tyc, and U. Leonhardt, New J. Phys. 13, 033016 (2011).
[CrossRef]

T. Tyc and X. Zhang, Nature 480, 42 (2011).
[CrossRef]

U. Leonhardt and S. Sahebdivan, J. Opt. 13, 024016 (2011).
[CrossRef]

2010 (2)

U. Leonhardt and T. G. Philbin, Phys. Rev. A 82, 057802 (2010).
[CrossRef]

T. D. Visser and E. Wolf, Opt. Commun. 283, 3371 (2010).
[CrossRef]

2009 (1)

U. Leonhardt, New J. Phys. 11, 093040 (2009).
[CrossRef]

2002 (1)

P. Saari, Laser Phys. 12, 812 (2002).

2001 (1)

1994 (1)

S. W. Hell, S. Lindek, and E. H. K. Stelzer, J. Mod. Opt. 41, 675 (1994).
[CrossRef]

1891 (1)

L. G. Gouy, Annal. Chim. Phys. 24, 145 (1891).

1890 (1)

L. G. Gouy, C. R. Hebd. Séances Acad. Sci. 110, 1251 (1890).

Arfken, G. B.

G. B. Arfken, H. J. Weber, and F. E. Harris, Mathematical Methods for Physicists (Harcourt, 2005).

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 2006).

Feng, S.

Gouy, L. G.

L. G. Gouy, Annal. Chim. Phys. 24, 145 (1891).

L. G. Gouy, C. R. Hebd. Séances Acad. Sci. 110, 1251 (1890).

Harris, F. E.

G. B. Arfken, H. J. Weber, and F. E. Harris, Mathematical Methods for Physicists (Harcourt, 2005).

Hell, S. W.

S. W. Hell, S. Lindek, and E. H. K. Stelzer, J. Mod. Opt. 41, 675 (1994).
[CrossRef]

Leonhardt, U.

U. Leonhardt and S. Sahebdivan, J. Opt. 13, 024016 (2011).
[CrossRef]

Y. G. Ma, S. Sahebdivan, C. K. Ong, T. Tyc, and U. Leonhardt, New J. Phys. 13, 033016 (2011).
[CrossRef]

U. Leonhardt and T. G. Philbin, Phys. Rev. A 82, 057802 (2010).
[CrossRef]

U. Leonhardt, New J. Phys. 11, 093040 (2009).
[CrossRef]

Lindek, S.

S. W. Hell, S. Lindek, and E. H. K. Stelzer, J. Mod. Opt. 41, 675 (1994).
[CrossRef]

Ma, Y. G.

Y. G. Ma, S. Sahebdivan, C. K. Ong, T. Tyc, and U. Leonhardt, New J. Phys. 13, 033016 (2011).
[CrossRef]

Ong, C. K.

Y. G. Ma, S. Sahebdivan, C. K. Ong, T. Tyc, and U. Leonhardt, New J. Phys. 13, 033016 (2011).
[CrossRef]

Philbin, T. G.

U. Leonhardt and T. G. Philbin, Phys. Rev. A 82, 057802 (2010).
[CrossRef]

Saari, P.

P. Saari, Laser Phys. 12, 812 (2002).

Sahebdivan, S.

Y. G. Ma, S. Sahebdivan, C. K. Ong, T. Tyc, and U. Leonhardt, New J. Phys. 13, 033016 (2011).
[CrossRef]

U. Leonhardt and S. Sahebdivan, J. Opt. 13, 024016 (2011).
[CrossRef]

Stelzer, E. H. K.

S. W. Hell, S. Lindek, and E. H. K. Stelzer, J. Mod. Opt. 41, 675 (1994).
[CrossRef]

Tyc, T.

Y. G. Ma, S. Sahebdivan, C. K. Ong, T. Tyc, and U. Leonhardt, New J. Phys. 13, 033016 (2011).
[CrossRef]

T. Tyc and X. Zhang, Nature 480, 42 (2011).
[CrossRef]

Visser, T. D.

T. D. Visser and E. Wolf, Opt. Commun. 283, 3371 (2010).
[CrossRef]

Weber, H. J.

G. B. Arfken, H. J. Weber, and F. E. Harris, Mathematical Methods for Physicists (Harcourt, 2005).

Winful, H. G.

Wolf, E.

T. D. Visser and E. Wolf, Opt. Commun. 283, 3371 (2010).
[CrossRef]

M. Born and E. Wolf, Principles of Optics (Cambridge University, 2006).

Zhang, X.

T. Tyc and X. Zhang, Nature 480, 42 (2011).
[CrossRef]

Annal. Chim. Phys. (1)

L. G. Gouy, Annal. Chim. Phys. 24, 145 (1891).

C. R. Hebd. Séances Acad. Sci. (1)

L. G. Gouy, C. R. Hebd. Séances Acad. Sci. 110, 1251 (1890).

J. Mod. Opt. (1)

S. W. Hell, S. Lindek, and E. H. K. Stelzer, J. Mod. Opt. 41, 675 (1994).
[CrossRef]

J. Opt. (1)

U. Leonhardt and S. Sahebdivan, J. Opt. 13, 024016 (2011).
[CrossRef]

Laser Phys. (1)

P. Saari, Laser Phys. 12, 812 (2002).

Nature (1)

T. Tyc and X. Zhang, Nature 480, 42 (2011).
[CrossRef]

New J. Phys. (2)

U. Leonhardt, New J. Phys. 11, 093040 (2009).
[CrossRef]

Y. G. Ma, S. Sahebdivan, C. K. Ong, T. Tyc, and U. Leonhardt, New J. Phys. 13, 033016 (2011).
[CrossRef]

Opt. Commun. (1)

T. D. Visser and E. Wolf, Opt. Commun. 283, 3371 (2010).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (1)

U. Leonhardt and T. G. Philbin, Phys. Rev. A 82, 057802 (2010).
[CrossRef]

Other (2)

M. Born and E. Wolf, Principles of Optics (Cambridge University, 2006).

G. B. Arfken, H. J. Weber, and F. E. Harris, Mathematical Methods for Physicists (Harcourt, 2005).

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Figures (3)

Fig. 1.
Fig. 1.

Comparison of the function sinckx (blue thick curve) and sinkx (red thin curve) describing a superposition of converging and diverging spherical waves and a plane wave, respectively, along an axis passing through the focal point. The phases match in the region x<0, while they differ by π for x>0, which demonstrates the Gouy phase shift.

Fig. 2.
Fig. 2.

Similar to Fig. 1, but for the 2D situation. The blue thick curve shows the cylindrical wave J0(kx), and the red thin curve shows the plane wave sin(kx+3π/4). The phases match for xλ=2π/k but differ by π/2 for xλ—a clear demonstration of the Gouy phase shift in two dimensions.

Fig. 3.
Fig. 3.

Propagation of the pulse described by Eq. (8). The sum was truncated at n=300 and smoothened by the Gaussian function exp[3.5(n/300)2] to eliminate rapid oscillations that would occur due to the truncation. The pulse is shown at times (a) t=0.2 (δ-like peak running toward the mirror), (b) t=1.2 (negative δ-like peak running back to the center), and (c) t=2.5 (a cotangentlike peak running toward the mirror). Arrows mark the direction of pulse propagation. The dramatic change of the peak shape after passing through the center is caused by the Gouy phase of π/2.

Equations (11)

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c2Δψψtt=0,
ψin=aexp[ikriωt]r,
ψ=ψin+ψout=2ikasinc(kr)exp(iωt).
J0(kx)2πkxcos(kxπ4).
ψ(r,t)=2rn=1sinknrsinknt.
ψ(r,t)=1rn=0{cos[kn(rt)]cos[kn(r+t)]},
=Δ(rt)Δ(r+t)r,
ψ(r,t)=n=12πknJ0(knr)sin(knt+π4).
kn(n14)π,
ψ(r,t)1rn=1{cos[kn(rt)]+sin[kn(r+t)]},
ψ(r,t)1rn=1{sin[kn(rτ)]cos[kn(r+τ)]}.

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